We demonstrate theoretically how the creation of polarization singularities by the evolution of a fractional nonuniform polarization optical element involves the peculiar mathematics of countably infinite sets in the form of "Hilbert's Hotel." Two distinct topological processes can be observed, depending on the structure of the fractional optical element.

We study the field that is produced by a paraxial refractive axicon lens. The results from geometrical optics, scalar wave optics, and electromagnetic diffraction theory are compared. In particular, the axial intensity, the on-axis effective wavelength, the transverse intensity, and the far-zone field are examined. A rigorous electromagnetic diffraction analysis shows that the state of polarization of the incident beam strongly affects the transverse intensity distribution, but not the intensity distribution in the far zone.

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We demonstrate that a J0-Bessel-correlated beam that is incident on a homogeneous sphere produces a highly unusual distribution of the scattered field, with the maximum no longer occurring in the forward direction. Such a beam can be easily generated using a spatially incoherent, annular source. Moreover, the direction of maximal scattering can be shifted by changing the spatial coherence length. In this process, the total power that is scattered remains constant. This new tool to control scattering directionality may be used to steer the scattered field away from the forward direction and selectively address detectors situated at different angles.

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